Multi-objective optimization problems (MOPs) with more than three objectives are called many-objective optimization problems (MaOPs) [1]}, [2]}. There are many MaOPs in real life [3]}, [4]}, [5]}, but traditional multi-objective evolutionary algorithms (MOEAs) based on domination and decomposition have difficulty in solving MaOPs [6]}. In contrast, indicator-based multi-objective optimization algorithms (IB-MOEAs) use indicator values to direct the search process when solving MaOPs. This method of evaluating indicators directly to search well-performing solutions is intuitive and gaining popularity. There are various indicators used in IB-MOEAs [7]}, such as hypervolume (HV) [8]}, generational distance (GD) [9]}, inverted generational distance (IGD) [10]}, epsilon indicator (\(\varepsilon \) -indicator) [11]}, shift-based density estimation (SDE) [12]}, R2 [13]}, \(\Delta _p\)  [14]} and so on.
